- Prof. Melvin Leok

Office: APM 5763

Email: mleok@math.ucsd.edu

Office Hours: MW 2:00pm-2:50pm, or by appointment.

- Course Handout
- The resources below are password protected with the user name ma277a,
and the password is the first 4 digits of:

.

- As there is no TA for this class, there will be no assessed homework.
- However, Prof. Darryl Holm, the author of one of the supplementary references, has a good collection of homework problems, and solutions: Homework Problems and Solutions
- You are encouraged to look over these problems, and try to work out some of these for yourself. If you would like suggestions for which ones to concentrate on, please contact Prof. Leok.

### Wednesday, June 10, 2015

3:00pm - 3:40pm Ali Behzadan Constrained Hamiltonian systems, gauge transformations, and general relativity 3:40pm - 4:00pm Jeremy Schmitt Hamilton-Jacobi theory and symplectic integrators 4:00pm - 4:20pm Poorya Mirkhosrav Multisymplectic discretization of the wave equation 4:30pm - 5:00pm Daniel Copeland The principle of gauge invariance in electrodynamics 5:00pm - 5:20pm Pun Tong Numerical calculation of the semi-classical limit of a Hamiltonian double-well system 5:20pm - 5:40pm Yiqun Li Variational integrators, Lie group methods and their applications 5:40pm - 6:00pm Joseph Palmer Variational principles for Lie-Poisson and Hamilton-Poincare equations 6:00pm - 6:30pm Alexander Kuczala The Hamilton-Jacobi-Bellman equation and dynamic programming

- This course will introduce geometric mechanics at the graduate level, which involves the use of geometric and symmetry techniques in the analysis of mechanical systems. In particular, we will discuss the variational principles and geometric structures that underly the Lagrangian and Hamiltonian formulation of mechanics, as well as introducing the relevant tools of differential geometry, such as manifolds, exterior calculus, and Lie groups. The course will culminate in a discussion of how symmetry and reduction theory serve as a unified basis for understanding the Eulerian description of rigid body dynamics and fluid mechanics.

- Some exposure to analytical mechanics is helpful, but not essential. The course will introduce the relevant differential geometric tools as well as the relevant aspects of analytical mechanics.

- Jerrold Marsden, Tudor Ratiu, Introduction to Mechanics and Symmetry, Second Edition, Springer-Verlag, 2002. ISBN: 038798643X. [ Electronic Version ]

- Taeyoung Lee, Melvin Leok, Harris McClamroch, Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds, Springer-Verlag, under contract. [ Draft Version ]
- Darryl Holm, Geometric Mechanics - Part I: Dynamics and Symmetry, Second Edition, Imperial College Press, 2011. ISBN: 184816775X. [ Electronic Version ]
- Ralph Abraham, Jerrold Marsden, Foundations of Mechanics, Second Edition, American Mathematical Society 2008. ISBN: 0821844385. [ Electronic Version ]
- Ralph Abraham, Jerrold Marsden, Tudor Ratiu, Manifolds, Tensor Analysis, and Applications, Second Edition, Springer-Verlag 1988. ISBN: 0387967907. [ Draft of Third Edition ]
- Vladimir Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Springer-Verlag 1989. ISBN: 0387968903. [ Electronic Version ]
- Anthony Bloch, Nonholonomic Mechanics and Control, Springer-Verlag, 2010. ISBN: 1441930434. [ Electronic Version ]
- Theodore Frankel, The Geometry of Physics, Second Edition, Cambridge University Press, 2003. ISBN: 0521539277. [ Electronic Version ]
- Darryl Holm, Geometric Mechanics - Part II: Rotating, Translating and Rolling, Second Edition, Imperial College Press, 2011. ISBN: 1848167784. [ First Edition ]
- Darryl Holm, Tanya Schmah, Cristina Stoica, Geometric Mechanics and Symmetry, Oxford University Press, 2009. ISBN: 0199212910. [ Electronic Version ]
- Chris Isham, Modern Differential Geometry for Physicits, Second Edition, World Scientific, 1999. ISBN: 9810235623. [ Electronic Version ]
- Jerrold Marsden, Lectures on Mechanics, Cambridge University Press, 1992. ISBN: 0521428440. [ Draft of Second Edition ]

- Your grade in the course is based on your project and your 20-minute project presentation during the time of the final.
- The topic of your project should be decided in consultation with the instructor before the end of the first month of class.